## Section2.4Group homomorphisms

### Subsection2.4.1

###### Definition2.4.1.Group homomorphism.
Let $G,H$ be groups.

A map $\phi\colon G\to H$ is called a homomorphism if

\begin{equation*} \phi(xy) = \phi(x)\phi(y) \end{equation*}

for all $x,y$ in $G\text{.}$ A homomorphism that is both injective (one-to-one) and surjective (onto) is called an isomorphism of groups. If $\phi\colon G\to H$ is an isomorphism, we say that $G$ is isomorphic to $H\text{,}$ and we write $G\approx H\text{.}$

Show that each of the following are homomorphisms.

• $GL(n,\R)\to \R^\ast$ given by $M\to \det M$
• $\Z\to \Z$ given by $x\to mx\text{,}$ some fixed $m\in \Z$
• $G\to G\text{,}$ $G$ any group, given by $x\to axa^{-1}\text{,}$ some fixed $a\in G$
• $\C^\ast\to\C^\ast$ given by $z\to z^2$

Show that each of the following are not homomorphisms. In each case, demonstrate what fails.

• $\Z\to \Z$ given by $x\to x+3$
• $\Z\to \Z$ given by $x\to x^2$
• $D_4\to D_4$ given by $g\to g^2$
###### Definition2.4.3.Kernel of a group homomorphism.

Let $\phi\colon G\to H$ be a group homomorphism, and let $e_H$ be the identity element for $H\text{.}$ We write $\ker(\phi)$ to denote the set

\begin{equation*} \ker(\phi) :=\phi^{-1}(e_H) = \{g\in G\colon \phi(g)=e_H\}, \end{equation*}

called the kernel of $\phi\text{.}$

Find the kernel of each of the following homomorphisms.

• $\C^\ast\to \C^\ast$ given by $z\to z^n$
• $\Z_8\to \Z_8$ given by $x\to 6x \pmod{8}$
• $G\to G\text{,}$ $G$ any group, given by $x\to axa^{-1}\text{,}$ some fixed $a\in G$
1. $\displaystyle C_n$
2. $\displaystyle \langle 4\rangle = \{0,4\}$
3. $\displaystyle \{x\in G\colon axa^{-1}=e\}=C(a)$

Here is a corollary of Proposition 2.4.6 and its proof.

###### Definition2.4.8.Normal subgroup.

A subgroup $H$ of a group $G$ is called normal if $ghg^{-1}\in H$ for every $g\in G\text{,}$ $h\in H\text{.}$ We write $H\trianglelefteq G$ to indicate that $H$ is a normal subgroup of $G\text{.}$

### Exercises2.4.2Exercises

###### 1.

Prove Properties 1 and 2.

###### 2.

Prove Properties 3 and 4.

###### 3.

Prove Properties 5, 6, and 7.

###### 4.
Show that the inverse of an isomorphism is an isomorphism.
###### 6.

Let $n,a$ be relatively prime positive integers. Show that the map $\Z_n\to \Z_n$ given by $x\to ax$ is an isomorphism.

Hint
Use the fact that $\gcd(m,n)$ is the least positive integer of the form $sm+tn$ over all integers $s,t$ (see Exercise 2.3.2.4). Use this to solve $ax=1 \pmod{n}$ when $a,n$ are relatively prime.
###### 7.Another construction of $\Z_n$.

Let $n\geq 1$ be an integer and let $\omega=e^{i2\pi/n}\text{.}$ Let $\phi\colon \Z\to S^1$ be given by $k\to \omega^k\text{.}$

1. Show that the the image of $\phi$ is the group $C_n$ of $n$th roots of unity.
2. Show that $\phi$ is a homomorphism, and that the kernel of $\phi$ is the set $n\Z=\{nk\colon k\in \Z\}\text{.}$
3. Conclude that $\Z/\!(n\Z)$ is isomorphic to the group of $n$-th roots of unity.
###### 8.Isomorphic images of generators are generators.

Let $S$ be a subset of a group $G\text{.}$ Let $\phi\colon G\to H$ be an isomorphism of groups, and let $\phi(S)=\{\phi(s)\colon s\in S\}\text{.}$ Show that $\phi(\langle S\rangle)=\langle \phi(S)\rangle\text{.}$

###### 9.Conjugation.

Let $G$ be a group, let $a$ be an element of $G\text{,}$ and let $C_a\colon G\to G$ be given by $C_a(g)=aga^{-1}\text{.}$ The map $C_a$ is called conjugation by the element $a$ and the elements $g,aga^{-1}$ are said to be conjugate to one another.

1. Show that $C_a$ is an isomorphism of $G$ with itself.

2. Show that "is conjugate to" is an equivalence relation. That is, consider the relation on $G$ given by $x\sim y$ if $y=C_a(x)$ for some $a\text{.}$ Show that this is an equivalence relation.

###### 10.Isomorphism induces an equivalence relation.

Prove that "is isomorphic to" is an equivalence relation on groups. That is, consider the relation $\approx$ on the set of all groups, given by $G\approx H$ if there exists a group isomorphism $\phi\colon G\to H\text{.}$ Show that this is an equivalence relation.

###### Characterization of normal subgroups.
Prove Proposition 2.4.9. That Item 1 is equivalent to Item 2 is established by Proposition 2.4.6.
###### 12.

Show that Item 3 implies Item 2. The messy part of this proof is to show that multiplication of cosets is well-defined. This means you start by supposing that $xK=x'K$ and $yK=y'K\text{,}$ then show that $xyK=x'y'K\text{.}$

###### 13.Further characterizations of normal subgroups.

Show that Item 3 is equivalent to the following conditions.

1. $gKg^{-1}= K$ for all $g\in G$
2. $gK = Kg$ for all $g\in G$
###### 14.Automorphisms.

Let $G$ be a group. An automorphism of $G$ is an isomorphism from $G$ to itself. The set of all automorphisms of $G$ is denoted $\Aut(G)$.

1. Show that $\Aut(G)$ is a group under the operation of function composition.

2. Show that

\begin{equation*} \Inn(G) := \{C_g\colon g\in G\} \end{equation*}

is a subgroup of $\Aut(G)\text{.}$ (The group $\Inn(G)$ is called the group of inner automorphisms of $G\text{.}$)

3. Find an example of an automorphism of a group that is not an inner automorphism.